From Thinking to Goals, From Thinking and Goals to Feedback and Hints

How many cubes are in the 10th step of this pattern? The 43rd step of this pattern? The nth step?

I’m interested in how a bit of theory about student thinking (proposed here) can help answer pressing teaching questions I’d have about this problem (listed here).

What is this problem good for? What math would students be able to learn from working on this problem?

Students aren’t going to learn about Shakespeare from working on this problem, because Shakespeare is unlikely to come up while working on this visual pattern. So, what thinking and ideas are likely to come up during this problem?

Our systematization of student thinking on visual patterns gives us (at least) a very good start, I think.

Roughly, there are three sorts of ways that students might think about this pattern:

  1. Recursively, seeing its growth in terms of the previous stages
  2. Relationally, connecting the step number to aspects of the shape at each stage
  3. Other students might reduce this pattern into numbers (essentially ignoring the shape) and then see the growth in recursive or relational terms (except now the relationship is an arithmetic one) [Example, here]

There’s also the business with expressing the generalized rule, no matter what it is or how it’s arrived at.

4. Students will either express a rule for the nth step in terms of algebra or in terms of instructions or steps for a calculation.

This, then, is roughly the math that my students will learn from working on, discussing and revising their work on this problem:

  1. They can learn to see this pattern (and others like it) recursively
  2. They can learn to see this pattern (and others like it) relationally (as in Kathryn’s lovely dialogue)
  3. They can learn to see this pattern (and others like it) only in terms of numbers and their growth or relationships
  4. They can learn how to express generalized rules in terms of algebra (or using some other sort of language)

That’s the learning I’m identifying that can happen from working on this problem, though I don’t claim to be exhaustive (more on that below). What potential learning does this analysis not include?

  • Contra Mimi, I don’t see think factorization and expansion, or other algebraic manipulation skills are likely to be improved while working on these problems. That’s not something that students are thinking about when they think about these problems.
  • Contra Tina, I don’t think that these problems will help students recognize patterns more globally, unless those patterns can be seen recursively or by relating the step number to aspects of the shape.

There are a lot of caveats that I need to add to the above. First, teachers are endlessly creative and I don’t doubt that Mimi and Tina’s classes worked on these skills. My point is that because algebraic manipulation and other sorts of pattern-reasoning don’t show up in most student thinking about finding “jump ahead” steps in visual patterns, some other prompt for thinking about those ideas has to come-in.

For example, Fawn does this. She tries to teach equivalence of expressions (i.e. factorization and expansion and like terms) via discussion after students have worked on these problems. So during discussion she prompts students to think about the equivalence of different people’s expressions, and that draws attention to thinking about algebraic manipulation. To my mind, this is a new call for thinking, one that is connected to the visual pattern but also substantively different from it.

Megan shares another great example of algebraic manipulation coming up from visual patterns. My only point is that kids won’t get better at algebraic manipulation from trying to find the 43rd or nth step.

Of course, there are lots of good reasons to do something in class besides helping kids learn something. Dan uses visual pattern problems to help foster a culture and expectation that students will work and figure out puzzles on their own. While I worry a bit about how expansive that culture and those expectations are (could a divide develop between opener problems and everything else that happens in class?) I think that this is certainly a legitimate goal that isn’t excluded by anything above.

One last word: I simplified the story and left out important parts. In the previous post I dug into some of the details about how kids use relational or recursive models for calculating a given step in the pattern. We can get into much more detail. A goal for a class or a student might be to relate the step number plus/minus some amount to a given step’s shape. A goal might be to calculate using multiplication (rather than addition) when using a recursive model. The more careful our analysis of how kids think about these questions, the sharper our goals can be.

What should I do if my students are stuck on this problem?

Part of what excites me about all this is that this analysis gives me a substantive way to work on the feedback and hints that I give. If we know how students think about a problem, we might get a sense of what ways of thinking a particular student could aspire to, and then we can think about hints and feedback strategically.

In particular, we can anticipate seeing various bits of the thinking that we outlined. We can identify productive next steps for students who are stuck, since we know where they are headed. Essentially, we can prepare a repertoire of hints or feedback to offer. Of course, we might adjust these hints/feedback on the fly given the specifics of context, but we can do significant preparation. (I’m not saying anything here that isn’t said better by Smith here.)

As illustrated by oodles of student work (again, as in Kathyrn’s post), students often see a pattern’s growth recursively and struggle to adopt a relational model. These students will likely express stuckness, since the “what’s the 43rd step?” question is laborious using the recursive model. (I also think it’s likely that a student with the recursive approach correctly infers that her methods are insufficient when she reads the call for the 43rd step, because why would a teacher ask you to do something simple and repetitive? That subtext is part of why a kid will report being stuck, even though they could add 2 43 times.)

What are questions, suggestions, hints, feedback, etc. that could help a student move from a recursive model to a relational one?

  • In the “Growing Worms” post I identified some middle-steps within recursive thinking that kids take on their way to a relational approach. This suggests that if a student is able to see the pattern recursively, we might benefit by nudging them towards multiplication.
    • “I’m hearing you say you could add 2 all these times, but that doesn’t feel right. I agree. What shortcuts can we come up with?”
  • What if a student uses multiplication, but fails to adjust or tinker? As a result, they’d end up using false proportional reasoning and likely have a false answer. This student likely wouldn’t feel stuck and wouldn’t ask for a hint. How can you help this student reach the “multiplies and tinkers” approach to calculating? (After all, that approach is right on the horizon for this student.)
    • “I notice that your answer is different than Dylan’s. Could you two chat about that?”
    • “One way that we can check a technique is by using it on really low numbers in a pattern. Can you show that your trick either does or doesn’t work for the low step numbers?”

And so on.

The big concern (raised on twitter in conversation with Lani Horn) is that classrooms, students and teaching situations are too different from each other for these sorts of feedback/hint suggestions to be useful.

I gain confidence from the big-girl and big-boy projects that, I think, this little one is aspiring to: Cognitively Guided Instruction [Fennema, Carpenter, Franke, Levi, Jacobs, & Empson (1996)] and the Math Assessment Resource Service ( These projects — especially CGI — make generalizations about student thinking that travel across classrooms. The MARS project offers suggestions for feedback that are specific to the task, but presumably grounded in their a learning progression that a kid might take.

(On the other hand, those are big-boy and big-girl projects done by lots of people over a very long time. An important question is how much a teacher, or even a few teachers working together, can reasonably expect to accomplish in this area.)

Ultimately the proof is in the pudding and I think I need to start working whatever this all this might turn into.

Next steps:

  • Go through a lot of student work on visual patterns. Make sure that we’re capturing the thinking that is going on.
  • Look at non-linear patterns. How is student thinking similar? How is it different?
  • What is something that this could turn into, to support teachers who are thinking of using visual patterns in their classes? An article? Maybe additions to A pdf guide, ala MARS’ lessons? A poster?
  • On the horizon: how would this generalize to function-finding, more generally? I know students look at tables of inputs/outputs in roughly similar ways — recursive, then relational once they have a model to relate outputs to inputs. Could all this be about a larger class of problems? And what sorts of problems are on the horizon after that? (Maybe this is as general as it’s worth going?)

2 thoughts on “From Thinking to Goals, From Thinking and Goals to Feedback and Hints

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